Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10,12,14)(11,13,15,16)(17,18,20,22)(19,23,24,21)(25,26,29,30)(27,31,34,37)(28,32,36,38)(33,39,35,40);(11,15)(13,16)(17,18,20,22)(19,21,24,23)(26,30)(31,37)(32,38)(39,40);(9,11)(10,13)(12,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(1,2)(3,6)(4,5)(7,8)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(1,3,4,7)(2,6,5,8)(9,10)(11,13)(12,14)(15,16)(17,19,20,24)(18,21,22,23)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=f,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4306435);